Singular Potentials and Regularization

Let $V\left(r\right)\mathrm{}$ be a potential repulsive and singular at the origin, and $A$ a physical quantity (say, the scattering length) associated with it. Let $V(r, \alpha )$ be a regulated version of the potential $V\left(r\right)\mathrm{}$; $V(r, \alpha )$ is nonsingular for $\alpha >0\mathrm{}$ and coincides with $V\left(r\right)\mathrm{}$ for $\alpha =0\mathrm{}$. Let $A\left(\alpha \right)$ be the corresponding scattering length. It is usually assumed that $A\left(0\right)=A$. Simple counterexamples are presented, namely, cases when $A$ and $A\left(0\right)\mathrm{}$, although both well defined and finite, are different. The existence of these counterexamples sheds doubt on the validity of a theorem given by Khuri and Pais. This doubt is substantiated by noting that the proof of the theorem contains an unjustified exchange of two limiting processes.